Abstract. We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced by an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper.

In the second part of this investigation are derived several forms of the bundle (analogue of the) Schrödinger equation governing the time evolution of state liftings of paths or sections along paths. We prove that up to a constant the matrix-bundle Hamiltonian, entering the bundle analogue of the matrix form of the conventional Schrödinger equation, coincides with the matrix of coefficients of the evolution transport. This allows us to interpret the Hamiltonian as a gauge field. We apply the bundle approach to the description of observables. It is shown that to any observable there corresponds a unique Hermitian lifting of paths or morphism along paths in corresponding bundles.

PACS numbers: 02.40.Ma, 03.65.Ca, 03.65.Ta, 04.40.Ma

Mathematics Subject Classification: 81P05, 81P99, 81Q99, 81S99

Print publication: Issue 23 (15 June 2001)
Received 28 October 2000, in final form 9 March 2001